Shapes of tight composite knots

Abstract

We present new computations of tight shapes obtained using the constrained gradient descent code ridgerunner for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of composite knots are at least 4π − 4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors. Our numerics support the connect sum conjecture and argue against the additivity of writhe conjecture. We also present data on the number of configurations having straight segments and highly curved kinked regions.